1. Closed loop, meaning you cant change anything cause it already happened. You time traveling was already written.

2. Back to the future, you can change things just be careful and dont accisently erase yourself through the butterfly effect.

3. We'll call this one reverse flash, not only can you change things but you can litterally kill your parents because you are disconnected from the timestream. (Not exclusive to number 2)

4. Future Trunks, congrats you just created a new timeline. You cannot change your future, have fun.

5. Alternate dimension, you just went to an alternate dimension that looks like your past. Dont let the British museum find out.

There's really only 2 not 5. Most are just variants of the two:

1. Singleton universe: time is linear, causality is set in stone, either through fix points (events that can't be changed). Time travel will just create a loop but causality dictates that this will already have happened and you can't change events.
2. Multiverse: You can go back in time, but doing so will branch the timeline and create a different universe. This will not have impact on any causality as you're an anomaly and you can do whatever you want (kill your grandpa, have a relationship with yourself etc.) and nothing will happen because you are in a parallel dimension and effectively can't produce paradoxes.

Sometimes movies just ignore paradoxes or logical fallacies because it makes for better screen time. But really all time travel is one of the two.

Primer is 1.
MCU is 2. (Endgame pretends to be 1 but really with the series Loki it's clear it's 2)
Predestination is 1.
The series Dark is actually a weird mix of the two. First it's 1. And then morphs into 2.
Triangle is 1.
Rick and Morty is 2.

I meant in fiction pedant. And there is one type of time travel, forward.